Many soaps feel greazy because, well, they are -being made from rendered animal fat. Some even tout moisturizing properties from lanolin which is just sheep goo akin to one's own hair grease.
I find vegetable based glycerin soap such as Pears does not suffer said greaziness and washes off cleanly sans residue feeling. Plus, it is not disgusting. So, I use that sparingly with a loofah pad or wash cloth ("face cloth") and scrub brush for nails. Fastidiously clean nails are a natural development of wearing contact lenses, if nothing else.
The advantage to cloths is they are sanitary (single-use before laundering) whereas loofahs are a bit dodgy and do not hold up well to laundering. Some cloths lack sufficient abrasiveness though so cheaper varieties may actually be better (Costco institutional bulk).
Those body parts besides the obvious dirty ones do not necessarily get soap every time -particularly the limbs. Ditto, hair gets shampoo as necessary -every alternate washing or so. The face gets a cetyl alcohol cleanser such as Cetaphil, rather than soap, to likewise avoid excessive drying. Admittedly, such cleanser does not "feel clean" if accustomed to soap so requires becoming reaccustomed unless already used to following with a moisturizer.
This is not a Patrick Bateman monologue.